Superconformal Ward Identities and their Solution

TL;DR

This paper derives superconformal Ward identities for four-point functions of BPS operators in four-dimensional superconformal theories, providing solutions that include various supermultiplets and analyzing crossing symmetry constraints.

Contribution

It introduces a method to solve superconformal Ward identities for four-point functions using harmonic polynomial expansions, accommodating different supermultiplet types.

Findings

Solutions include long, short, and semi-short supermultiplets with protected or free dimensions.

Dependence on R-symmetry and conformal invariants is simplified via null vectors and polynomial expansions.

Crossing symmetry conditions are analyzed within the derived framework.

Abstract

Superconformal Ward identities are derived for the the four point functions of chiral primary BPS operators for $\N=2,4$ superconformal symmetry in four dimensions. Manipulations of arbitrary tensorial fields are simplified by introducing a null vector so that the four point functions depend on two internal $R$-symmetry invariants as well as two conformal invariants. The solutions of these identities are interpreted in terms of the operator product expansion and are shown to accommodate long supermultiplets with free scale dimensions and also short and semi-short multiplets with protected dimensions. The decomposition into $R$-symmetry representations is achieved by an expansion in terms of two variable harmonic polynomials which can be expressed also in terms of Legendre polynomials. Crossing symmetry conditions on the four point functions are also discussed.